The runoff will tend to creep under the hoods and can be a big problem in itself unless the bike is kept pointed sharply downhill during the flush out.

So do pay attention to this, and keep a bucket under the levers or the oil will hit the ground and splash all over rims, tires, shoes, etc.

Outside I put the bike against the railing and lift the rear wheel a couple of feet upward, then secure the wheel and bike to the upper run of the railing.
I actually reposition the bike to the other side before spraying out the other lever (if needed), because I want un obstructed access to the lever with the spray.

And, if the lever still works when it is warm, or if the lever is moved Veeeery slowly, then there is NO need to use the WD40.
Simply spray out the innards with a foaming oil product like shown below, which contains enough solvent to creep into the fine annular spaces about the pawl pivot pins.
Allow the lever to drip out for 20 minutes, and have a towel wedged up between the levers to soak up the settling oil. Stuff towel under the levers also to pull excess oil out from around the lever pivot spring.
It is not unheard of for the oil to make it's way into a cable housing, which can draw dirt into the housing and affect shifting quality. So be aware if any shifting hesitation is noted afterward.